[Proof of Proposition  REF]
Emden–Fowler equation deals with a differential equation with a form of \(\frac{d^2 \varphi }{dq^2} = A q^n \varphi ^m\) , and the differential equation that we have corresponds to the case of \(A=-1\) , \(n=1\) , and \(m=-1/2\) .
In this case, the solutions can be written in parametric form [1]}:
for constants \(a\)  and \(b\)  such that \(A = -\frac{9}{2} (b/a)^3\) ,
\( q(\theta ) = a \theta ^{-\frac{2}{3}}\left[ \left( \theta Z^{\prime }(\theta ) + \frac{1}{3} Z(\theta ) \right)^2 - \theta ^2 Z^2(\theta ) \right], ~\varphi (\theta ) = b \theta ^{\frac{2}{3}} Z^2(\theta ), ~Z(\theta ) = C_1 I_{1/3}(\theta ) + C_2 K_{1/3}(\theta ),\) 
